Limits at Infinity

So let’s consider a function defined for , where this interval means the set . It really doesn’t matter here what is, just that we’ve got some point where is defined for all larger numbers. We want to come up with a sensible definition for .

When we took a limit at a point we said that if for every there is a so that implies . But this talk of and is all designed to stand in for neighborhoods in a metric space. Picking a defines a neighborhood of the point . All we need is to come up with a notion of a “neighborhood” of .

What we’ll use is a ray just like the one above: . This seems to make sense as the collection of real numbers “near” infinity. So let’s drop it into our definition: the limit of a function at infinity, is if for every there is an so that implies . It’s straightforward to verify from here that this definition of limit satisfies the same laws of limits as the earlier definition.

Finally, we can define neighborhoods of as leftward rays . Then we get a similar definition of the limit of a function at .

One particular limit that’s useful to have as a starting point is . Indeed, given we can set . Then if we see that , establishing the limit.

From here we can handle the limit at infinity of any rational function . Let’s split off the top degree terms from the polynomials and . Divide through top and bottom by to write

Now every term in has degree less than , so each is a multiple of some power of . The laws of limits then tell us that they go to , and the limit of the denominator of is . Thus our limit is the limit of the numerator.

If we have a positive power of as our leading term, which goes up to or down to (depending on the sign of . If , all the powers are negative, and thus the limit is . And if , then all the other powers are negative, and the limit is .

So if the numerator of has the higher degree, we have . If the denominator has higher degree, then . If the degrees are equal, we compare the leading coefficients and find .

[…] bigger, trying to fill out the whole ray . And for each one we have a value for the integral: . So we take the limit as approaches infinity: . This will be the value of the integral over the entire […]

[…] the function is then nondecreasing, and a nondecreasing function bounded above must have a finite limit at infinity. Indeed, the set of values of must be bounded above, and so there is a least upper bound . […]

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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